Well, there are lots of questions that you asked, but let's go one by one:
When I say it's not a good way of using globals, or better to say when someone say that your coding style isn't that beautiful, it doesn't mean that it doesn't work. It only means that it could have been better. When you're using a function, you should pass variables to it (using arguments), but in here, you're not passing anything to your function and instead using global variables. Also you're using $GET which is a super global array, it means you don't even need to use the global keywork in a function or class that it can recognize it, these super global variables are defined in the global scope, it means wherever you call them, you can get them. Then you're using $REMOTE_ADDR which is pretty old, instead you can use $SERVER["REMOTE_ADDR"] and then if you wana use it, you don't need to call that global keyword either.
But if it's me, I wouldn't even use a function for sessioning a person in here, because it's no need that you do that! You write a function when you wana use it in multiple places, but in here, you only use your function once in the login script. So, it's better that in the login page you check for the username and password, then if it's right, you session your user and also write his info in your session table. Then in each page you should check to see if the user is sessioned, and if not, you should send him back to your login page. Then you should delete all the records from the db that are older than 3 minutes, and then try to select the user in the table, if you can not find him, it means he's expired and you should redirect him to the login page again. I also think that using 3 minutes as your time limit isn't enough, you should put 15 or 30 minutes at least, how many pages should your visitor see in only 3 minutes?! Ok, then you should update the time of your user. Don't forget that you should do all these things in all your pages. So, You can put them in a file and include it in all your pages.
There is something else that I forgot to mention, you are NOT using the super global $SESSION array, which means you can not access your sessions in your starting_session function unless you have:
global $ENTRYS;
So, my suggestion is that you totally use the $_SESSION array and do not use these deprecated functions.
It's also a good idea to write that Updating line alone, means not in any if, that you see if it works. If it doesn't then it's something wrong with your DB (that I really do not think it's your case), but if it does, then you should do the changes that I said in your code and tell us what happend in your journey, ok?!
Happy coding,
bijan